Lines $L_1, L_2, .., L_10 $are distinct among which the lines $L_2, L_4, L_6, L_8, L_{10}$ are parallel to each other and the lines $L_1, L_3, L_5, L_7, L_9$ pass through a given point C. The number of point of intersection of pairs of lines from the complete set $L_1, L_2, L_3, ..., L_{10}$ is 

Total Number of Intersection Points

Given:

• 10 distinct lines: \( L_1, L_2, \ldots, L_{10} \)
• \( L_2, L_4, L_6, L_8, L_{10} \): parallel (no intersections among them)
• \( L_1, L_3, L_5, L_7, L_9 \): concurrent at point \( C \) (intersect at one point)

? Calculation:

\[ \text{Total line pairs: } \binom{10}{2} = 45 \]

\[ \text{Subtract parallel pairs: } \binom{5}{2} = 10 \Rightarrow 45 - 10 = 35 \]

\[ \text{Concurrent at one point: reduce } 10 \text{ pairs to 1 point} \Rightarrow 35 - 9 = \boxed{26} \]

✅ Final Answer: \(\boxed{26}\) unique points of intersection